Shifted Legendre approach to the analysis and identification of a linear delayed system with a nonlinear gain

Applications of the shifted Legendre polynomials expansion to the analysis and identification of the nonlinear time-delayed system, described by a memoryless nonlinear element followed by a linear plant with time delay, are studied. The system described here is assumed both controllable and observable. For analysis, by using the shifted Legendre polynomials expansion, the solution of a nonlinear state equation is reduced to the solution of a linear algebraic matrix equation. For identification, through the shifted Legendre expansions of the measured input/output data, the unknown parameters of both the linear delayed plant and the characterisation of the nonlinear element are estimated by using the least-squares method. Algorithms are presented. Numerical examples are given to illustrate the use of this approach.